Finite Group Databasebeta

← #100769

Subgroups — #100769

Order 512.

462 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28449C211yesmaximal, characteristic
2#23353C211yesmaximal, characteristic
3#9905C212yesmaximal, complemented
4#8579C211yesmaximal, characteristic
5#7049C212yesmaximal, complemented
6C2×#497—21no—
7C2×#462—21no—
8C2×#461—21no—
9#2868C2211yescharacteristic
10C4×#124C2211yescharacteristic
11#2016C412yes—
12#2013C412yes—
13#1985—21no—
14#1960—21no—
15#1924—22no—
16#1870—21no—
17#1851—22no—
18#1831C2212yes—
19#1767C2212yes—
20#1767C2211yescharacteristic
21C2×#329—21no—
22#1296—22no—
23#1296—22no—
24#1295—21no—
25C22×Q16—22no—
26C22×SD16—22no—
27C22×M16—21no—
28C2×C4⋊Q8—21no—
29C2×C4⋊Q8—21no—
30C2×C4⋊Q8C2×C412yes—
31C2×#124D411yescharacteristic
32C2×C4×Q8D411yescharacteristic
33C2×C4×D4D411yescharacteristic
34#497—44no—
35#462—42no—
36#462—42no—
37#461—42no—
38#461—42no—
39C2×C4⋊C8—21no—
40C2×C4⋊C8—22no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8D412yes—
43C2×C4⋊C8D412yes—
44C2×#103—22no—
45C2×#103—22no—
46C2×C22⋊C8—22no—
47C2×C22⋊C8—21no—
48C2×#97—21no—
49C2×#97—22no—
50C2×#97—21no—
51#391C2×C411yescharacteristic
52#390C2×C411yescharacteristic
53#388—22no—
54#384—22no—
55#384—22no—
56#382D411yescharacteristic
57C4×C22⋊C4—22no—
58C43C2311yescharacteristic
59#336—22no—
60#336—22no—
61#336—22no—
62#336—22no—
63#336C2×C412yes—
64#329—42no—
65C22×Q8—22no—
66C22×Q8C22⋊C411yescharacteristic
67C22×D4C22⋊C411yescharacteristic
68C23×C4—21no—
69C2×Q16—44no—
70C2×Q16—44no—
71C2×Q16—82no—
72C2×SD16—44no—
73C2×SD16—44no—
74C2×SD16—82no—
75C2×M16—81no—
76C2×M16—82no—
77C22×C8—22no—
78C22×C8—22no—
79C22×C8—22no—
80C22×C8—42no—
81C4⋊Q8—41no—
82C4⋊Q8—41no—
83C4⋊Q8—41no—
84C4⋊Q8—41no—
85C4⋊Q8—41no—
86C4⋊Q8—42no—
87C4⋊Q8—41no—
88C4⋊Q8—82no—
89#124—21no—
90#124—22no—
91#124—21no—
92C4×Q8—41no—
93C4×Q8—41no—
94C4×D4—41no—
95C4×D4—41no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—24no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4C22⋊C411yescharacteristic
102C2×C4⋊C4C22⋊C411yescharacteristic
103C2×C22⋊C4—22no—
104C2×C22⋊C4—22no—
105C2×C22⋊C4—21no—
106C2×C42—21no—
107C2×C42C22×C411yescharacteristic
108C2×C42C2×D411yescharacteristic
109C2×C42C2×D412yes—
110C2×C42C2×D411yescharacteristic
111C2×C42#4112yes—
112C4⋊C8—22no—
113C4⋊C8—22no—
114C4⋊C8—24no—
115C4⋊C8—42no—
116C4⋊C8—42no—
117C4⋊C8—42no—
118C4⋊C8—81no—
119#103—44no—
120#103—44no—
121C22⋊C8—81no—
122C22⋊C8—82no—
123#97—42no—
124#97—42no—
125#97—42no—
126#97—81no—
127#95—22no—
128#95—22no—
129C24—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—41no—
133C2×Q8—41no—
134C2×Q8—42no—
135C2×Q8—42no—
136C2×Q8—42no—
137C2×Q8—42no—
138C2×Q8—42no—
139C2×Q8—42no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—41no—
143C2×D4—41no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4C2×C22⋊C411yescharacteristic
153C22×C4C4×D412yes—
154C22×C4#12011yescharacteristic
155C22×C4#12112yes—
156C22×C4#12311yescharacteristic
157Q16—88no—
158SD16—88no—
159M16—84no—
160C2×C8—24no—
161C2×C8—42no—
162C2×C8—42no—
163C2×C8—44no—
164C2×C8—42no—
165C2×C8—42no—
166C2×C8—42no—
167C2×C8—42no—
168C2×C8—42no—
169C2×C8—42no—
170C2×C8—82no—
171C2×C8—82no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—81no—
175C4⋊C4—84no—
176C4⋊C4—81no—
177C4⋊C4—81no—
178C4⋊C4—81no—
179C4⋊C4—81no—
180C4⋊C4—82no—
181C4⋊C4—82no—
182C22⋊C4—42no—
183C22⋊C4—42no—
184C22⋊C4—82no—
185C22⋊C4—81no—
186C42—21no—
187C42—21no—
188C42—22no—
189C42—22no—
190C42—41no—
191C42—42no—
192C42—41no—
193C42—41no—
194C42AGL(1,ℤ/8)11yescharacteristic
195C42AGL(1,ℤ/8)12yes—
196C42#13711yescharacteristic
197C23—42no—
198C23—41no—
199C23—41no—
200C23—42no—
201C23—41no—
202C23#38611yescharacteristic
203Q8—42no—
204Q8—81no—
205Q8—82no—
206Q8—82no—
207Q8—82no—
208Q8—82no—
209D4—42no—
210D4—81no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—42no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—81no—
228C2×C4—81no—
229C2×C4—82no—
230C2×C4—82no—
231C2×C4—82no—
232C2×C4—82no—
233C2×C4#41711yescharacteristic
234C2×C4#41811yescharacteristic
235C2×C4#45311yescharacteristic
236C2×C4#45311yescharacteristic
237C2×C4#45512yes—
238C2×C4#45612yes—
239C2×C4#46812yes—
240C2×C4#46912yes—
241C8—44no—
242C8—84no—
243C8—82no—
244C8—82no—
245C22—81no—
246C22—82no—
247C22—82no—
248C22—81no—
249C22—81no—
250C22#188212yes—
251C22#188711yescharacteristic
252C22#189511yescharacteristic
253C22#190212yes—
254C22#191411yescharacteristic
255C4—22no—
256C4—22no—
257C4—21no—
258C4—21no—
259C4—42no—
260C4—41no—
261C4—81no—
262C4—81no—
263C4—81no—
264C4—162no—
265C4#165911yescharacteristic
266C4#166011yescharacteristic
267C4#166112yes—
268C2—161no—
269C2#900311yescharacteristic
270C2#932012yes—
271C2#933712yes—
272C2#940511yescharacteristic
273C2#942111yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
prime radicalC22×D4C22⋊C4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #9905C2semidirect
5. #7049C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#284492
#233532
#99052
#85792
#70492